Logic Gates

Step 1: Interpreting the given statement.
The statement “Mangoes are delicious but expensive” can be written logically as: where : Mangoes are delicious, and : Mangoes are expensive.
Step 2: Understanding the concept of duality.
The dual of a statement is obtained by interchanging logical AND ( ) with logical OR ( ), without negating the propositions.
Step 3: Writing the dual statement.
The dual of is That is, “Mangoes are delicious or Mangoes are expensive.”
Step 4: Conclusion.
Hence, the correct dual statement is option (C).

To find the negation of , we will apply De Morgan's laws. 1. First, apply the negation to the entire expression: According to De Morgan's law, the negation of a conjunction is the disjunction of the negations: 2. Now, apply De Morgan’s law to and : - - Thus, the negation of the original expression becomes: 3. Simplifying this further, we get the final result:

Let's analyze the given statement . This statement is a biconditional statement which is false only when one side is true and the other is false. First, consider as true and as false. Then will be true, so becomes true true, which is true. Thus, true and false doesn't make the given statement false. Next, consider as false and as true. Then will be false, so becomes false false, which is true. Therefore, false and true doesn't make the given statement false either. Now, let's consider as false and as false. Then will be true. Therefore, becomes false true, which is false. This satisfies the condition of the question. Now, let's check the options for as false and as false: A. is false. B. becomes false (false true), which is false true = true. C. becomes false (true false) = false. D. becomes (false true) false = true false = false. Option B is the only statement which is true when is false and is false.

The given Boolean expression is: First, simplify the second part of the expression, . By the distributive property: Since , we can simplify this further: Now, the original expression becomes: Simplify this expression by combining like terms: Finally, by the absorption law, , so the expression reduces to: Thus, the final simplified expression is: Thus, the correct answer is .

A tautology is a statement that is always true, regardless of the truth values of its components. The statement will be a tautology if is equivalent to . Let's analyze the antecedent: . This is only true if is true and is true. If is true, then must be false. If is false and is true, then must also be false (because if were true, would be false). Thus, the antecedent is only true when both and are false. The entire statement is a conditional statement which is only false when the antecedent is true and the consequent is false. To make it a tautology, must be true whenever is true, which is equivalent to being true. Therefore, must be equivalent to .